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text.actor {
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.note {
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/* Today line */
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  stroke: #E83737;
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  stroke-width: 2; }

.doneCrit0,
.doneCrit1,
.doneCrit2,
.doneCrit3 {
  stroke: #E83737;
  fill: lightgrey;
  stroke-width: 2;
  cursor: pointer;
  shape-rendering: crispEdges; }

.milestone {
  transform: rotate(45deg) scale(0.8, 0.8); }

.milestoneText {
  font-style: italic; }

.doneCritText0,
.doneCritText1,
.doneCritText2,
.doneCritText3 {
  fill: #323D47 !important; }

.activeCritText0,
.activeCritText1,
.activeCritText2,
.activeCritText3 {
  fill: #323D47 !important; }

.titleText {
  text-anchor: middle;
  font-size: 18px;
  fill: #323D47;
   }

g.classGroup text {
  fill: #9370DB;
  stroke: none;
  
  font-size: 10px; }
  g.classGroup text .title {
    font-weight: bolder; }

g.classGroup rect {
  fill: #BDD5EA;
  stroke: #9370DB; }

g.classGroup line {
  stroke: #9370DB;
  stroke-width: 1; }

.classLabel .box {
  stroke: none;
  stroke-width: 0;
  fill: #BDD5EA;
  opacity: 0.5; }

.classLabel .label {
  fill: #9370DB;
  font-size: 10px; }

.relation {
  stroke: #9370DB;
  stroke-width: 1;
  fill: none; }

#compositionStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#compositionEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#aggregationStart {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1; }

#aggregationEnd {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1; }

#dependencyStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#dependencyEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#extensionStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#extensionEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

.commit-id,
.commit-msg,
.branch-label {
  fill: lightgrey;
  color: lightgrey;
   }

.pieTitleText {
  text-anchor: middle;
  font-size: 25px;
  fill: #eee;
}

g.stateGroup text {
  stroke: none;
  font-size: 10px;
}

g.stateGroup circle {
  fill: white !important;
  stroke: white !important;
}

g.stateGroup .state-title {
  font-weight: bolder;
  fill: black; }

g.stateGroup rect {
  fill: #ececff;
  stroke: #9370DB; }

g.stateGroup line {
  stroke: #9370DB;
  stroke-width: 1; }

.transition {
  stroke: #9370DB;
  stroke-width: 1;
  fill: none; }

.stateGroup .composit {
  fill: #555;
  border-bottom: 1px; }

.state-note {
  stroke: rgba(255, 255, 255, 0.25);
  fill: #fff5ad; }
  .state-note text {
    fill: black;
    stroke: none;
    font-size: 10px; }

.stateLabel .box {
  stroke: none;
  stroke-width: 0;
  fill: #BDD5EA;
  opacity: 0.5; }

.stateLabel text {
  fill: black;
  font-size: 10px;
  font-weight: bold;
}

.cluster-label {
  color:black;
}

.statediagram-cluster rect {
  fill: #BDD5EA;
  stroke: #9370DB; 
  stroke-width: 1px;
}
.statediagram-cluster rect.outer {
  rx: 5px;
  ry: 5px;
}
.statediagram-state .divider {
  stroke: #9370DB; 
}

.statediagram-state .title-state {
  rx: 5px;
  ry: 5px;
}
.statediagram-cluster.statediagram-cluster .inner {
  fill: white;
}
.statediagram-cluster.statediagram-cluster-alt .inner {
  fill: #e0e0e0;
}

.statediagram-cluster .inner {
  rx:0;
  ry:0;
}

.statediagram-state rect.basic {
  rx: 5px;
  ry: 5px;
}
.statediagram-state rect.divider {
  stroke-dasharray: 10,10;
  fill: #efefef;
}

.note-edge {
  stroke-dasharray: 5;
}

.statediagram-note rect {
  stroke: var(--cluster-border);
  fill: #fff5ad;
  stroke-width: 1px;
  rx: 0;
  ry: 0;
}

.node circle.state-start {
  fill: black;
  stroke: black;
}
.node circle.state-end {
  fill: black;
  stroke: white;
  stroke-width: 1.5
}
#statediagram-barbEnd {
  fill: #9370DB; 
}

/* CSS Document */

/** code highlight */

.cm-s-inner .cm-variable,
.cm-s-inner .cm-operator,
.cm-s-inner .cm-property {
    color: #b8bfc6;
}

.cm-s-inner .cm-keyword {
    color: #C88FD0;
}

.cm-s-inner .cm-tag {
    color: #7DF46A;
}

.cm-s-inner .cm-attribute {
    color: #7575E4;
}

.CodeMirror div.CodeMirror-cursor {
    border-left: 1px solid #b8bfc6;
    z-index: 3;
}

.cm-s-inner .cm-string {
    color: #D26B6B;
}

.cm-s-inner .cm-comment,
.cm-s-inner.cm-comment {
    color: #DA924A;
}

.cm-s-inner .cm-header,
.cm-s-inner .cm-def,
.cm-s-inner.cm-header,
.cm-s-inner.cm-def {
    color: #8d8df0;
}

.cm-s-inner .cm-quote,
.cm-s-inner.cm-quote {
    color: #57ac57;
}

.cm-s-inner .cm-hr {
    color: #d8d5d5;
}

.cm-s-inner .cm-link {
    color: #d3d3ef;
}

.cm-s-inner .cm-negative {
    color: #d95050;
}

.cm-s-inner .cm-positive {
    color: #50e650;
}

.cm-s-inner .cm-string-2 {
    color: #f50;
}

.cm-s-inner .cm-meta,
.cm-s-inner .cm-qualifier {
    color: #b7b3b3;
}

.cm-s-inner .cm-builtin {
    color: #f3b3f8;
}

.cm-s-inner .cm-bracket {
    color: #997;
}

.cm-s-inner .cm-atom,
.cm-s-inner.cm-atom {
    color: #84B6CB;
}

.cm-s-inner .cm-number {
    color: #64AB8F;
}

.cm-s-inner .cm-variable {
    color: #b8bfc6;
}

.cm-s-inner .cm-variable-2 {
    color: #9FBAD5;
}

.cm-s-inner .cm-variable-3 {
    color: #1cc685;
}

.CodeMirror-selectedtext,
.CodeMirror-selected {
    background: #4a89dc;
    color: #fff !important;
    text-shadow: none;
}

.CodeMirror-gutters {
    border-right: none;
}

/* CSS Document */

/** markdown source **/
.cm-s-typora-default .cm-header, 
.cm-s-typora-default .cm-property
{
    color: #cebcca;
}

.CodeMirror.cm-s-typora-default div.CodeMirror-cursor{
    border-left: 3px solid #b8bfc6;
}

.cm-s-typora-default .cm-comment {
    color: #9FB1FF;
}

.cm-s-typora-default .cm-string {
    color: #A7A7D9
}

.cm-s-typora-default .cm-atom, .cm-s-typora-default .cm-number {
    color: #848695;
    font-style: italic;
}

.cm-s-typora-default .cm-link {
    color: #95B94B;
}

.cm-s-typora-default .CodeMirror-activeline-background {
    background: rgba(51, 51, 51, 0.72);
}

.cm-s-typora-default .cm-comment, .cm-s-typora-default .cm-code {
	color: #8aa1e1;
}@import "";
@import "";
@import "";

:root {
    --bg-color:  #363B40;
    --side-bar-bg-color: #2E3033;
    --text-color: #b8bfc6;

    --select-text-bg-color:#4a89dc;

    --item-hover-bg-color: #0a0d16;
    --control-text-color: #b7b7b7;
    --control-text-hover-color: #eee;
    --window-border: 1px solid #555;

    --active-file-bg-color: rgb(34, 34, 34);
    --active-file-border-color: #8d8df0;

    --primary-color: #a3d5fe;

    --active-file-text-color: white;
    --item-hover-bg-color: #70717d;
    --item-hover-text-color: white;
    --primary-color: #6dc1e7;

    --rawblock-edit-panel-bd: #333;

    --search-select-bg-color: #428bca;
}

html {
    font-size: 16px;
}

html,
body {
    -webkit-text-size-adjust: 100%;
    -ms-text-size-adjust: 100%;
    background: #363B40;
    background: var(--bg-color);
    fill: currentColor;
    line-height: 1.625rem;
}

#write {
    max-width: 1080px;
}


@media only screen and (min-width: 1400px) {
	#write {
		max-width: 1024px;
	}
}

@media only screen and (min-width: 1800px) {
	#write {
		max-width: 1200px;
	}
}

html,
body,
button,
input,
select,
textarea,
div.code-tooltip-content {
    color: #b8bfc6;
    border-color: transparent;
}

div.code-tooltip,
.md-hover-tip .md-arrow:after {
    background: #333;
}

.popover.bottom > .arrow:after {
    border-bottom-color: #333;
}

html,
body,
button,
input,
select,
textarea {
    font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}

hr {
    height: 2px;
    border: 0;
    margin: 24px 0 !important;
}

h1,
h2,
h3,
h4,
h5,
h6 {
    font-family: "Lucida Grande", "Corbel", sans-serif;
    font-weight: normal;
    clear: both;
    -ms-word-wrap: break-word;
    word-wrap: break-word;
    margin: 0;
    padding: 0;
    color: #DEDEDE
}

h1 {
    font-size: 2.5rem;
    /* 36px */
    line-height: 2.75rem;
    /* 40px */
    margin-bottom: 1.5rem;
    /* 24px */
    letter-spacing: -1.5px;
}

h2 {
    font-size: 1.63rem;
    /* 24px */
    line-height: 1.875rem;
    /* 30px */
    margin-bottom: 1.5rem;
    /* 24px */
    letter-spacing: -1px;
    font-weight: bold;
}

h3 {
    font-size: 1.17rem;
    /* 18px */
    line-height: 1.5rem;
    /* 24px */
    margin-bottom: 1.5rem;
    /* 24px */
    letter-spacing: -1px;
    font-weight: bold;
}

h4 {
    font-size: 1.12rem;
    /* 16px */
    line-height: 1.375rem;
    /* 22px */
    margin-bottom: 1.5rem;
    /* 24px */
    color: white;
}

h5 {
    font-size: 0.97rem;
    /* 16px */
    line-height: 1.25rem;
    /* 22px */
    margin-bottom: 1.5rem;
    /* 24px */
    font-weight: bold;
}

h6 {
    font-size: 0.93rem;
    /* 16px */
    line-height: 1rem;
    /* 16px */
    margin-bottom: 0.75rem;
    color: white;
}

@media (min-width: 980px) {
    h3.md-focus:before,
    h4.md-focus:before,
    h5.md-focus:before,
    h6.md-focus:before {
        color: #ddd;
        border: 1px solid #ddd;
        border-radius: 3px;
        position: absolute;
        left: -1.642857143rem;
        top: .357142857rem;
        float: left;
        font-size: 9px;
        padding-left: 2px;
        padding-right: 2px;
        vertical-align: bottom;
        font-weight: normal;
        line-height: normal;
    }

    h3.md-focus:before {
        content: 'h3';
    }

    h4.md-focus:before {
        content: 'h4';
    }

    h5.md-focus:before {
        content: 'h5';
        top: 0px;
    }

    h6.md-focus:before {
        content: 'h6';
        top: 0px;
    }
}

a {
    text-decoration: none;
    outline: 0;
}

a:hover {
    outline: 0;
}

a:focus {
    outline: thin dotted;
}

sup.md-footnote {
    background-color: #555;
    color: #ddd;
}

p {
    -ms-word-wrap: break-word;
    word-wrap: break-word;
}

p,
ul,
dd,
ol,
hr,
address,
pre,
table,
iframe,
.wp-caption,
.wp-audio-shortcode,
.wp-video-shortcode {
    margin-top: 0;
    margin-bottom: 1.5rem;
    /* 24px */
}

li > blockquote {
	margin-bottom: 0;
}

audio:not([controls]) {
    display: none;
}

[hidden] {
    display: none;
}

::-moz-selection {
    background: #4a89dc;
    color: #fff;
    text-shadow: none;
}

*.in-text-selection,
::selection {
    background: #4a89dc;
    color: #fff;
    text-shadow: none;
}

ul,
ol {
    padding: 0 0 0 1.875rem;
    /* 30px */
}

ul {
    list-style: square;
}

ol {
    list-style: decimal;
}

ul ul,
ol ol,
ul ol,
ol ul {
    margin: 0;
}

b,
th,
dt,
strong {
    font-weight: bold;
}

i,
em,
dfn,
cite {
    font-style: italic;
}

blockquote {
    padding-left: 1.875rem;
    margin: 0 0 1.875rem 1.875rem;
    border-left: solid 2px #474d54;
    padding-left: 30px;
    margin-top: 35px;
}

pre,
code,
kbd,
tt,
var {
    font-size: 0.875rem;
    font-family: Monaco, Consolas, "Andale Mono", "DejaVu Sans Mono", monospace;
}

code,
tt,
var {
    background: rgba(0, 0, 0, 0.05);
}

kbd {
    padding: 2px 4px;
    font-size: 90%;
    color: #fff;
    background-color: #333;
    border-radius: 3px;
    box-shadow: inset 0 -1px 0 rgba(0,0,0,.25);
}

pre.md-fences {
    padding: 10px 10px 10px 30px;
    margin-bottom: 20px;
    background: #333;
}

.CodeMirror-gutters {
    background: #333;
    border-right: 1px solid transparent;
}

.enable-diagrams pre.md-fences[lang="sequence"] .code-tooltip,
.enable-diagrams pre.md-fences[lang="flow"] .code-tooltip,
.enable-diagrams pre.md-fences[lang="mermaid"] .code-tooltip {
    bottom: -2.2em;
    right: 4px;
}

code,
kbd,
tt,
var {
    padding: 2px 5px;
}

table {
    max-width: 100%;
    width: 100%;
    border-collapse: collapse;
    border-spacing: 0;
}

th,
td {
    padding: 5px 10px;
    vertical-align: top;
}

a {
    -webkit-transition: all .2s ease-in-out;
    transition: all .2s ease-in-out;
}

hr {
    background: #474d54;
    /* variable */
}

h1 {
    margin-top: 2em;
}

a {
    color: #e0e0e0;
    text-decoration: underline;
}

a:hover {
    color: #fff;
}

.md-inline-math script {
    color: #81b1db;
}

b,
th,
dt,
strong {
    color: #DEDEDE;
    /* variable */
}

mark {
    background: #D3D40E;
}

blockquote {
    color: #9DA2A6;
}

table a {
    color: #DEDEDE;
    /* variable */
}

th,
td {
    border: solid 1px #474d54;
    /* variable */
}

.task-list {
    padding-left: 0;
}

.md-task-list-item {
    padding-left: 1.25rem;
}

.md-task-list-item > input {
    top: auto;
}

.md-task-list-item > input:before {
    content: "";
    display: inline-block;
    width: 0.875rem;
    height: 0.875rem;
    vertical-align: middle;
    text-align: center;
    border: 1px solid #b8bfc6;
    background-color: #363B40;
    margin-top: -0.4rem;
}

.md-task-list-item > input:checked:before,
.md-task-list-item > input[checked]:before {
    content: '\221A';
    /*◘*/
    font-size: 0.625rem;
    line-height: 0.625rem;
    color: #DEDEDE;
}

/** quick open **/
.auto-suggest-container {
    border: 0px;
    background-color: #525C65;
}

#typora-quick-open {
    background-color: #525C65;
}

#typora-quick-open input{
    background-color: #525C65;
    border: 0;
    border-bottom: 1px solid grey;
}

.typora-quick-open-item {
    background-color: inherit;
    color: inherit;
}

.typora-quick-open-item.active,
.typora-quick-open-item:hover {
    background-color: #4D8BDB;
    color: white;
}

.typora-quick-open-item:hover {
    background-color: rgba(77, 139, 219, 0.8);
}

.typora-search-spinner > div {
  background-color: #fff;
}

#write pre.md-meta-block {
    border-bottom: 1px dashed #ccc;
    background: transparent;
    padding-bottom: 0.6em;
    line-height: 1.6em;
}

.btn,
.btn .btn-default {
    background: transparent;
    color: #b8bfc6;
}

.ty-table-edit {
    border-top: 1px solid gray;
    background-color: #363B40;
}

.popover-title {
    background: transparent;
}

.md-image>.md-meta {
    color: #BBBBBB;
    background: transparent;
}

.md-expand.md-image>.md-meta {
    color: #DDD;
}

#write>h3:before,
#write>h4:before,
#write>h5:before,
#write>h6:before {
    border: none;
    border-radius: 0px;
    color: #888;
    text-decoration: underline;
    left: -1.4rem;
    top: 0.2rem;
}

#write>h3.md-focus:before {
    top: 2px;
}

#write>h4.md-focus:before {
    top: 2px;
}

.md-toc-item {
    color: #A8C2DC;
}

#write div.md-toc-tooltip {
    background-color: #363B40;
}

.dropdown-menu .btn:hover,
.dropdown-menu .btn:focus,
.md-toc .btn:hover,
.md-toc .btn:focus {
    color: white;
    background: black;
}

#toc-dropmenu {
    background: rgba(50, 54, 59, 0.93);
    border: 1px solid rgba(253, 253, 253, 0.15);
}

#toc-dropmenu .divider {
    background-color: #9b9b9b;
}

.outline-expander:before {
    top: 2px;
}

#typora-sidebar {
    box-shadow: none;
    border-right: 1px dashed;
    border-right: none;
}

.sidebar-tabs {
    border-bottom:0;
}

#typora-sidebar:hover .outline-title-wrapper {
    border-left: 1px dashed;
}

.outline-title-wrapper .btn {
    color: inherit;
}

.outline-item:hover {
    border-color: #363B40;
    background-color: #363B40;
    color: white;
}

h1.md-focus .md-attr,
h2.md-focus .md-attr,
h3.md-focus .md-attr,
h4.md-focus .md-attr,
h5.md-focus .md-attr,
h6.md-focus .md-attr,
.md-header-span .md-attr {
    color: #8C8E92;
    display: inline;
}

.md-comment {
    color: #5a95e3;
    opacity: 1;
}

.md-inline-math svg {
    color: #b8bfc6;
}

#math-inline-preview .md-arrow:after {
    background: black;
}

.modal-content {
    background: var(--bg-color);
    border: 0;
}

.modal-title {
    font-size: 1.5em;
}

.modal-content input {
    background-color: rgba(26, 21, 21, 0.51);
    color: white;
}

.modal-content .input-group-addon {
    color: white;
}

.modal-backdrop {
    background-color: rgba(174, 174, 174, 0.7);
}

.modal-content .btn-primary {
    border-color: var(--primary-color);
}

.md-table-resize-popover {
    background-color: #333;
}

.form-inline .input-group .input-group-addon {
    color: white;
}

#md-searchpanel {
    border-bottom: 1px dashed grey;
}

/** UI for electron */

.context-menu,
#spell-check-panel,
#footer-word-count-info {
    background-color: #42464A;
}

.context-menu.dropdown-menu .divider,
.dropdown-menu .divider {
    background-color: #777777;
}

footer {
    color: inherit;
}

@media (max-width: 1000px) {
    footer {
        border-top: none;
    }
    footer:hover {
        color: inherit;
    }
}

#file-info-file-path .file-info-field-value:hover {
    background-color: #555;
    color: #dedede;
}

.megamenu-content,
.megamenu-opened header {
    background: var(--bg-color);
}

.megamenu-menu-panel h2,
.megamenu-menu-panel h1,
.long-btn {
    color: inherit;
}

.megamenu-menu-panel input[type='text'] {
    background: inherit;
    border: 0;
    border-bottom: 1px solid;
}

#recent-file-panel-action-btn {
    background: inherit;
    border: 1px grey solid;
}

.megamenu-menu-panel .dropdown-menu > li > a {
    color: inherit;
    background-color: #2F353A;
    text-decoration: none;
}

.megamenu-menu-panel table td:nth-child(1) {
    color: inherit;
    font-weight: bold;
}

.megamenu-menu-panel tbody tr:hover td:nth-child(1) {
    color: white;
}

.modal-footer .btn-default, 
.modal-footer .btn-primary,
.modal-footer .btn-default:not(:hover) {
    border: 1px solid;
    border-color: transparent;
}

.btn-default:hover, .btn-default:focus, .btn-default.focus, .btn-default:active, .btn-default.active, .open > .dropdown-toggle.btn-default {
    color: white;
    border: 1px solid #ddd;
    background-color: inherit;
}

.modal-header {
    border-bottom: 0;
}

.modal-footer {
    border-top: 0;
}

#recent-file-panel tbody tr:nth-child(2n-1) {
    background-color: transparent !important;
}

.megamenu-menu-panel tbody tr:hover td:nth-child(2) {
    color: inherit;
}

.megamenu-menu-panel .btn {
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<div id='write'  class=''><h2><a name="第四章回合更新monte-carlo-update）价值迭代" class="md-header-anchor"></a><span>第四章：回合更新（Monte Carlo update）价值迭代</span></h2><p><span>本章开始介绍无模型的强化学习算法。无模型强化学习算法在没有环境的数学描述的情况下，只依靠经验（例如轨迹样本）学习出给定策略的价值函数和最优策略。</span></p><p><span>本章介绍的回合更新算法只能用于回合制任务，它在每个回合结束后更新价值函数，与有模型迭代更新类似，也是先学习策略评估，再学习最优策略求解。回合更新的策略评估的基本思路是用 Monte Carlo 方法来估计价值函数的期望值，所以回合更新的英文为 Monte Carlo update 。</span></p><p><span>由第二章内容可以知道在有模型的情况下，借助于动力 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.259ex" height="1.746ex" viewBox="-39 -500.4 542 751.6" role="img" focusable="false" style="vertical-align: -0.583ex; margin-left: -0.091ex;"><defs><path stroke-width="0" id="E407-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E407-MJMATHI-70" x="0" y="0"></use></g></svg></span><script type="math/tex">p</script><span> 的表达式，状态价值函数可以表示动作价值函数；借助于策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E461-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E461-MJMATHI-3C0" x="0" y="0"></use></g></svg></span><script type="math/tex">\pi</script><span> 的表达式，动作价值函数可以表示状态价值函数。但对于无模型情况下，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.259ex" 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stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E407-MJMATHI-70" x="0" y="0"></use></g></svg></span><script type="math/tex">p</script><span> 的表达式是未知的，因此不能用状态价值表示动作价值，只能用动作价值表示状态价值；而策略改进可仅由动作价值函数确定，故动作价值函数往往更加重要。</span></p><p><span>在同一个回合中，同一个状态（或状态动作对）可能会被多次访问；若采用回合内全部的样本值更新价值函数，则称为</span><strong><span>每次访问回合更新</span></strong><span>（every visit Monte Carlo update）；若每个回合只采用第一次访问的样本更新价值函数，则称为</span><strong><span>首次访问回合更新</span></strong><span>（first visit Monte Carlo update）；它们都能收敛到真实的价值函数。</span></p><p><span>回合更新算法包括同策回合更新算法和异策回合更新算法。同策回合更新算法是指采样（生成轨迹）的策略和被评估或被优化的策略是同一个策略；异策回合更新算法则允许两者的策略是不同的策略。</span></p><h3><a name="一同策回合更新" class="md-header-anchor"></a><span>一、同策回合更新</span></h3><p><span>同策的每次访问回合更新策略评估算法，即可以评估策略的状态价值；也可以评估策略的动作价值，然后使用 Bellman 期望方程求得状态价值。同策的每次访问回合更新策略评估算法如下：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n9" cid="n9" mdtype="math_block"><div 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transform="translate(0,-9791)"><g><rect fill="black" stroke="none" width="41597" height="100" x="0" y="-500"></rect></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-391">\; \\ \; \\
\large \textbf{算法 4-1   每次访问回合更新评估策略的动作（或状态）价值} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：环境（无数学描述） ，策略 $\pi$ 。} \\
&\text{输出：价值函数 $q(s,a),\; s \in \mathcal S, a \in \mathcal A$，（或 $v(s),\; s \in \mathcal S$ ）。} \\
&\text{1.（初始化）初始化价值估计 $q(s,a) \leftarrow$ 任意值，$s \in \mathcal S, a \in \mathcal A$，（或 $v(s) \leftarrow$ 任意值，$s \in \mathcal S$）。若更} \\
&\qquad \text{新价值需要使用计数器，则初始化计数器 $c(s,a) \leftarrow 0,\; s \in \mathcal S, a \in \mathcal A$，（或 $c(s) \leftarrow 0,\; s \in \mathcal S$）。} \\
&\text{2.（回合更新）对于每个回合执行以下操作：} \\
&\qquad \text{2.1（采样）用策略 $\pi$ 生成轨迹 $S_0,A_0,R_1,S_1,\cdots,S_{T-1},A_{T-1},R_T,S_T$ 。} \\
&\qquad \text{2.2（初始化回报）$G \leftarrow 0$ 。} \\
&\qquad \text{2.3（逐步更新）对 $t \leftarrow T-1,T-2,\cdots,0$ ，执行以下步骤：} \\
&\qquad \qquad \text{2.3.1（更新回报）$G \leftarrow \gamma G + R_{t+1}$ ；} \\
&\qquad \qquad \text{2.3.2（更新价值）更新 $q(S_t,A_t)$（或 $v(S_t)$ ）以减小 $[G-q(S_t,A_t)]^2$（或 $[G-v(S_t)]^2$ ）。} \\
&\qquad \qquad \qquad \;\, \text{如 $c(S_t,A_t) \leftarrow c(S_t,A_t)+1$ ，$q(S_t,A_t) \leftarrow q(S_t,A_t) + \frac{1}{c(S_t,A_t)}[G-q(S_t,A_t)]$} \\
&\qquad \qquad \qquad \;\, \text{（或 $c(S_t) \leftarrow c(S_t)+1$ ，$v(S_t) \leftarrow v(S_t) + \frac{1}{c(S_t)}[G-v(S_t)]$ ）。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>与有模型迭代更新类似，算法 4-1 中可以用参数来控制回合更新的回合数，例如，使用最大回合数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.692ex" height="2.228ex" viewBox="0 -749.6 2020.4 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E408-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 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140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E408-MJMATHI-6B" x="0" y="0"></use><g transform="translate(521,-150)"><use transform="scale(0.707)" xlink:href="#E408-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMATHI-78" x="1406" y="0"></use></g></g></svg></span><script type="math/tex">k_{max}</script><span> 或者精度指标 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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给出了同策的首次访问回合更新策略评估算法，相对于算法 4-1 不同点在于，在每次得到轨迹样本后，需要先找出各个状态分别在哪些步骤首次访问，然后在后续更新过程中，只在那些首次访问的步骤更新价值函数的估计值。</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n13" cid="n13" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-392-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="42.067ex" viewBox="-18.1 -43.5 43612.5 18112.2" role="img" focusable="false" style="vertical-align: -41.966ex; margin-left: -0.042ex; max-width: 100%;"><defs><path stroke-width="0" id="E394-MJMAINB-34" d="M531 0Q510 3 381 3Q238 3 214 0H201V62H313V155H32V217L205 434Q342 606 362 630T387 655L391 656Q395 656 401 656T414 656H427Q447 656 451 645Q453 641 453 429V217H542V155H453V62H542V0H531ZM324 217V494L103 218L213 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\large \textbf{算法 4-2   首次访问回合更新评估策略的动作（或状态）价值} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\cdots \quad \text{同算法 4-1 } \quad \cdots \\
&\text{2.3（初始化首次出现的步骤）$f(s,a) \leftarrow -1,\; s \in \mathcal S, a \in \mathcal A$，（或 $f(s) \leftarrow -1,\; s \in \mathcal S$ ）。} \\
&\text{2.4（统计首次出现的步骤数）对于 $t \leftarrow 0,1,\cdots,T-1$ ，执行以下步骤：} \\
&\qquad \text{如果 $f(S_t,A_t) < 0$ ，则 $f(S_t,A_t) \leftarrow t$ ，（或 $f(S_t) < 0$ ，则 $f(S_t) \leftarrow t$ ）。} \\
&\text{2.5（逐步更新）对 $t \leftarrow T-1,T-2,\cdots,0$ ，执行以下步骤：} \\
&\qquad \text{2.5.1（更新回报）$G \leftarrow \gamma G + R_{t+1}$ ；} \\
&\qquad \text{2.5.2（首次出现则更新）如果 $f(S_t,A_t)=t$（或 $f(S_t)=t$ ），则更新价值函数，} \\
&\qquad \qquad \;\, \text{以减小 $[G-q(S_t,A_t)]^2$（或 $[G-v(S_t)]^2$ ）。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>在算法 4-1 和算法 4-2 的更新价值估计后，进行策略改进，那么就会得到新的策略，这样不断更新，就有希望找到最优策略，这就是同策回合更新的基本思想。</span></p><p><span>但如果只是简单地将回合更新策略评估的算法移植为同策回合更新算法，时长会困于局部最优而找不到全局最优策略。例如同策算法可能会从一个并不好的策略出发，只经过那些较优的状态，然后只为那些较优的状态更新动作价值，而那些最优状态的动作价值没有更新仍然为 0；基于这些动作价值更新策略，只会使得策略去寻找较优的状态，而无法寻找最优的状态；更新后的策略依旧只选择那些较优的状态，反复如此，就陷入了局部最优。</span></p><p><span>为解决以上问题，提出了</span><strong><span>起始探索</span></strong><span>（exploring start）这一概念，让所有可能的状态动作对都成为可能的回合起点，这样就不会遗漏任何状态动作对，但是在理论上，目前并不清楚该带起始探索的同策回合更新算法是否总能收敛到最优策略。</span></p><p><span>带起始探索的回合更新算法也有每次访问和首次访问，算法 4-3 给出了每次访问的算法，首次访问的算法可以参照算法 4-2 在算法 4-3 的基础上修改得到，此处略。</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n18" cid="n18" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-393-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; 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\large \textbf{算法 4-3   带起始探索的每次访问同策回合更新算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{1.（初始化）初始化动作价值估计 $q(s,a) \leftarrow$ 任意值，$s \in \mathcal S, a \in \mathcal A$ 。若更新价值需要使用计数器，} \\
&\qquad \text{则初始化计数器 $c(s,a) \leftarrow 0,\; s \in \mathcal S, a \in \mathcal A$ 。} \\
&\text{2.（回合更新）对于每个回合执行以下操作：} \\
&\qquad \text{2.1（起始探索）选择 $S_0 \in \mathcal S, A_0 \in \mathcal A$ ，使得每一个状态动作对都可能被选为 $(S_0,A_0)$ 。} \\
&\qquad \text{2.2（采样）用策略 $\pi$ 生成轨迹 $S_0,A_0,R_1,S_1,\cdots,S_{T-1},A_{T-1},R_T,S_T$ 。} \\
&\qquad \text{2.3（初始化回报）$G \leftarrow 0$ 。} \\
&\qquad \text{2.4（逐步更新）对 $t \leftarrow T-1,T-2,\cdots,0$ ，执行以下步骤：} \\
&\qquad \qquad \text{2.4.1（更新回报）$G \leftarrow \gamma G + R_{t+1}$ ；} \\
&\qquad \qquad \text{2.4.2（更新动作价值）更新 $q(S_t,A_t)$ 以减小 $[G-q(S_t,A_t)]^2$。} \\
&\qquad \qquad \text{2.4.3（策略改进）$\pi(S_t) \leftarrow \underset{a}{\arg\max} \; q(S_t,a)$（若有多个 $a$ 取到最大值，则任选一个）。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>在很多环境中，并不是任意状态都能作为回合的起始状态，例如电动游戏往往有着相对固定的起始状态，因此需要新的办法来解决陷入局部最优的问题——</span><strong><span>基于柔性策略的回合更新算法</span></strong><span>。</span></p><p><strong><span>柔性策略（soft policy）：</span></strong><span>对于某个策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E461-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 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y="0"></use></g></svg></span><script type="math/tex">\pi(a \mid s)>0</script><span> 。因此柔性策略理论上是可以覆盖所有的状态或状态动作对的。</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.395ex" height="1.649ex" viewBox="0 -625 600.7 710.1" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E423-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g><use transform="scale(1.2)" xlink:href="#E423-MJMATHI-3B5" x="0" y="0"></use></g><use transform="scale(1.2)" xlink:href="#E423-MJMATHI-3B5" x="34" y="0"></use></g></svg></span><script type="math/tex">\large{\pmb{\varepsilon}}</script><span> </span><strong><span>柔性策略（</span></strong><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.395ex" height="1.649ex" viewBox="0 -625 600.7 710.1" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E423-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 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x="2793" y="0"></use><use xlink:href="#E421-MJMAIN-3E" x="3460" y="0"></use><g transform="translate(4238,0)"><g transform="translate(397,0)"><rect stroke="none" width="2742" height="60" x="0" y="220"></rect><use xlink:href="#E421-MJMATHI-3B5" x="1138" y="676"></use><g transform="translate(60,-694)"><use xlink:href="#E421-MJMAIN-7C" x="0" y="0"></use><use xlink:href="#E421-MJCAL-41" x="278" y="0"></use><use xlink:href="#E421-MJMAIN-28" x="1097" y="0"></use><use xlink:href="#E421-MJMATHI-73" x="1486" y="0"></use><use xlink:href="#E421-MJMAIN-29" x="1955" y="0"></use><use xlink:href="#E421-MJMAIN-7C" x="2344" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\displaystyle \pi(a \mid s) > \frac{\varepsilon}{|\mathcal A(s)|}</script><span> 。对于给定的环境上的某个确定性策略，在所有的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.082ex" height="1.36ex" viewBox="0 -500.4 466 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E427-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E427-MJMATHI-3B5" x="0" y="0"></use></g></svg></span><script type="math/tex">\varepsilon</script><span> 柔性策略中有一个策略最接近这个确定性策略，这个策略称为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.395ex" height="1.649ex" viewBox="0 -625 600.7 710.1" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E423-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g><use transform="scale(1.2)" xlink:href="#E423-MJMATHI-3B5" x="0" y="0"></use></g><use transform="scale(1.2)" xlink:href="#E423-MJMATHI-3B5" x="34" y="0"></use></g></svg></span><script type="math/tex">\large{\pmb{\varepsilon}}</script><strong><span>贪心策略</span></strong><span>（</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.082ex" height="1.36ex" viewBox="0 -500.4 466 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E427-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E427-MJMATHI-3B5" x="0" y="0"></use></g></svg></span><script type="math/tex">\varepsilon</script><span> - greedy policy）。具体而言，对于确定性策略：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n22" cid="n22" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-394-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="5.797ex" viewBox="0 -1497.2 42321.7 2496" role="img" focusable="false" style="vertical-align: -2.32ex; max-width: 100%;"><defs><path stroke-width="0" id="E396-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E396-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 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y="0"></use><use xlink:href="#E396-MJMATHI-61" x="962" y="0"></use><use xlink:href="#E396-MJMAIN-2223" x="1768" y="0"></use><use xlink:href="#E396-MJMATHI-73" x="2324" y="0"></use><use xlink:href="#E396-MJMAIN-29" x="2793" y="0"></use><use xlink:href="#E396-MJMAIN-3D" x="3460" y="0"></use><g transform="translate(4516,0)"><use xlink:href="#E396-MJSZ3-7B"></use><g transform="translate(917,0)"><g transform="translate(-19,0)"><g transform="translate(0,557)"><use xlink:href="#E396-MJMAIN-31" x="0" y="0"></use><use xlink:href="#E396-MJMAIN-2C" x="500" y="0"></use><use xlink:href="#E396-MJMATHI-61" x="1944" y="0"></use><use xlink:href="#E396-MJMAIN-3D" x="2751" y="0"></use><g transform="translate(3807,0)"><use xlink:href="#E396-MJMATHI-61" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E396-MJMAIN-2217" x="748" y="513"></use></g></g><g transform="translate(0,-643)"><use xlink:href="#E396-MJMAIN-30" x="0" y="0"></use><use xlink:href="#E396-MJMAIN-2C" x="500" y="0"></use><use xlink:href="#E396-MJMATHI-61" x="1944" y="0"></use><use xlink:href="#E396-MJMAIN-2260" x="2751" y="0"></use><g transform="translate(3807,0)"><use xlink:href="#E396-MJMATHI-61" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E396-MJMAIN-2217" x="748" y="513"></use></g></g></g></g></g><use xlink:href="#E396-MJMATHI-73" x="12538" y="0"></use><use xlink:href="#E396-MJMAIN-2208" x="13285" y="0"></use><use xlink:href="#E396-MJCAL-53" x="14230" y="0"></use><use xlink:href="#E396-MJMAIN-2C" x="14872" y="0"></use><use xlink:href="#E396-MJMATHI-61" x="15316" y="0"></use><use xlink:href="#E396-MJMAIN-2208" x="16123" y="0"></use><use xlink:href="#E396-MJCAL-41" x="17068" y="0"></use><use xlink:href="#E396-MJMAIN-28" x="17887" y="0"></use><use xlink:href="#E396-MJMATHI-73" x="18276" y="0"></use><use xlink:href="#E396-MJMAIN-29" x="18745" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-394">\pi(a \mid s) = 
\begin{cases}
1, \quad a = a^* \\
0, \quad a \ne a^*
\end{cases}
\qquad s \in \mathcal S, a \in \mathcal A(s)</script></div></div><p><span>对应的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.082ex" height="1.36ex" viewBox="0 -500.4 466 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E427-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E427-MJMATHI-3B5" x="0" y="0"></use></g></svg></span><script type="math/tex">\varepsilon</script><span> 贪心策略是：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n24" cid="n24" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-395-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="10.427ex" viewBox="0 -2494 42321.7 4489.5" role="img" focusable="false" style="vertical-align: -4.635ex; max-width: 100%;"><defs><path stroke-width="0" id="E397-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" 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y="0"></use><use xlink:href="#E397-MJMAIN-2208" x="22188" y="0"></use><use xlink:href="#E397-MJCAL-41" x="23133" y="0"></use><use xlink:href="#E397-MJMAIN-28" x="23952" y="0"></use><use xlink:href="#E397-MJMATHI-73" x="24341" y="0"></use><use xlink:href="#E397-MJMAIN-29" x="24810" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-395">\pi(a \mid s) = 
\begin{cases}
\displaystyle 1 - \varepsilon + \frac{\varepsilon}{|\mathcal A(s)|}, \quad a = a^* \\
\displaystyle \frac{\varepsilon}{|\mathcal A(s)|}, \qquad \qquad \;\, a \ne a^*
\end{cases}
\qquad s \in \mathcal S, a \in \mathcal A(s)</script></div></div><p><span>用 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.082ex" height="1.36ex" viewBox="0 -500.4 466 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E427-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E427-MJMATHI-3B5" x="0" y="0"></use></g></svg></span><script type="math/tex">\varepsilon</script><span> 贪心策略的表达式更新策略仍然满足策略改进定理。（证明：略）</span></p><p><span>算法 4-4 给出了基于柔性策略的每次访问同策回合更新算法，基于首次访问的算法同理可以参照前面的算法来修改算法 4-4 得到，此处略。值得一提的是，算法中策略改进的操作不一定要在每步更新价值函数后就立即进行，当回合步数较长，但是状态较少时，可以在价值函数完全更新完毕后统一进行；或者也可以不显式维护策略，而是在采样过程中决定动作时用动作价值函数隐含的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.082ex" height="1.36ex" viewBox="0 -500.4 466 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E427-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 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\large \textbf{算法 4-4   基于柔性策略的每次访问同策回合更新算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{1.（初始化）初始化动作价值估计 $q(s,a) \leftarrow$ 任意值，$s \in \mathcal S, a \in \mathcal A$ 。若更新价值需要使用计数器，} \\
&\qquad \text{则初始化计数器 $c(s,a) \leftarrow 0,\; s \in \mathcal S, a \in \mathcal A$ ；初始化策略 $\pi(\cdot \mid \cdot)$ 为任意 $\varepsilon$ 柔性策略。} \\
&\text{2.（回合更新）对于每个回合执行以下操作：} \\
&\qquad \text{2.1（采样）用策略 $\pi$ 生成轨迹 $S_0,A_0,R_1,S_1,\cdots,S_{T-1},A_{T-1},R_T,S_T$ 。} \\
&\qquad \text{2.2（初始化回报）$G \leftarrow 0$ 。} \\
&\qquad \text{2.3（逐步更新）对 $t \leftarrow T-1,T-2,\cdots,0$ ，执行以下步骤：} \\
&\qquad \qquad \text{2.3.1（更新回报）$G \leftarrow \gamma G + R_{t+1}$ ；} \\
&\qquad \qquad \text{2.3.2（更新动作价值）更新 $q(S_t,A_t)$ 以减小 $[G-q(S_t,A_t)]^2$。} \\
&\qquad \qquad \text{2.3.3（策略改进）$A^* \leftarrow \underset{a}{\arg\max} \; q(S_t,a)$ ，更新策略 $\pi(\cdot \mid S_t)$ 为贪心策略} \\
&\qquad \qquad \qquad \;\, \text{$\pi(a \mid S_t)=0,\; a \ne A^*$ 对应的 $\varepsilon$ 柔性策略。如 $\pi(a \mid S_t) \leftarrow \frac{\varepsilon}{|\mathcal A(s)|},\; a\in \mathcal A(s)$ ，} \\
&\qquad \qquad \qquad \;\, \text{$\pi(A^* \mid S_t) \leftarrow \pi(A^* \mid S_t)+(1-\varepsilon)$ 。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><h3><a name="二异策回合更新" class="md-header-anchor"></a><span>二、异策回合更新</span></h3><p><span>在统计学上，</span><strong><span>重要性采样</span></strong><span>（importance sampling）是一种用一个分布生成的样本来估计另一个分布的统计量的方法。异策回合更新算法就是基于重要性采样的：在异策学习中，将要学习的策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E461-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 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x="1060" y="-213"></use></g><use xlink:href="#E399-MJMAIN-29" x="17990" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-397">\begin{split}
Pr_\pi&[A_t,R_{t+1},S_{t+1},\cdots,S_{T-1},A_{T-1},R_T,S_T \mid S_t] \\
&=\pi(A_t \mid S_t) \, p(S_{t+1},R_{t+1} \mid S_t,A_t) \, \pi(A_{t+1} \mid S_{t+1}) \cdots p(S_T,R_T \mid S_{T-1},A_{T-1}) \\
&=\prod_{\tau=t}^{T-1} \pi(A_\tau \mid S_\tau) \prod_{\tau=t}^{T-1}p(S_{\tau+1},R_{\tau+1}\mid S_\tau,A_\tau) \\ \\

Pr_b&[A_t,R_{t+1},S_{t+1},\cdots,S_{T-1},A_{T-1},R_T,S_T \mid S_t] \\
&=b(A_t \mid S_t) \, p(S_{t+1},R_{t+1} \mid S_t,A_t) \, b(A_{t+1} \mid S_{t+1}) \cdots p(S_T,R_T \mid S_{T-1},A_{T-1}) \\
&=\prod_{\tau=t}^{T-1} b(A_\tau \mid S_\tau) \prod_{\tau=t}^{T-1}p(S_{\tau+1},R_{\tau+1}\mid S_\tau,A_\tau) \\
\end{split}</script></div></div><p><span>这两个概率的比值定义为</span><strong><span>重要性采样比率</span></strong><span>（importance sample ratio）：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n34" cid="n34" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-398-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="6.955ex" viewBox="0 -1746.4 42321.7 2994.3" role="img" focusable="false" style="vertical-align: -2.784ex; margin-bottom: -0.115ex; max-width: 100%;"><defs><path stroke-width="0" id="E400-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 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xlink:href="#E443-MJSZ2-2211" x="0" y="0"></use><g transform="translate(148,-1088)"><use transform="scale(0.707)" xlink:href="#E443-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E443-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E443-MJMAIN-31" x="1123" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E443-MJMATHI-63" x="804" y="1626"></use></g><g transform="translate(2637,0)"><use xlink:href="#E443-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E443-MJMATHI-69" x="674" y="-213"></use></g></g></svg></span><script type="math/tex">\displaystyle \frac{1}{c} \sum_{i=1}^c g_i</script><span> 来作为价值函数的估计，这样的方法实际上默认了这 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.006ex" height="1.36ex" viewBox="0 -500.4 433 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path 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style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E461-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E461-MJMATHI-3C0" x="0" y="0"></use></g></svg></span><script type="math/tex">\pi</script><span> 而言，各样本的出现概率是各轨迹的重要性采样比率，这样我们可以用加权平均来完成  Monte Carlo 估计。具体而言，若 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.615ex" 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transform="scale(0.707)" xlink:href="#E448-MJMATHI-69" x="731" y="-213"></use><use xlink:href="#E448-MJMAIN-28" x="1138" y="0"></use><use xlink:href="#E448-MJMAIN-31" x="1527" y="0"></use><use xlink:href="#E448-MJMAIN-2264" x="2305" y="0"></use><use xlink:href="#E448-MJMATHI-69" x="3361" y="0"></use><use xlink:href="#E448-MJMAIN-2264" x="3984" y="0"></use><use xlink:href="#E448-MJMATHI-63" x="5039" y="0"></use><use xlink:href="#E448-MJMAIN-29" x="5472" y="0"></use></g></svg></span><script type="math/tex">\rho_i \; (1 \le i \le c)</script><span> 是回报样本 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.907ex" height="1.746ex" viewBox="0 -500.4 821 751.6" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E453-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 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transform="translate(148,-1088)"><use transform="scale(0.707)" xlink:href="#E450-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E450-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E450-MJMAIN-31" x="1123" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E450-MJMATHI-63" x="804" y="1626"></use><g transform="translate(1610,0)"><use xlink:href="#E450-MJMATHI-3C1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E450-MJMATHI-69" x="731" y="-213"></use></g></g></g></g></svg></span><script type="math/tex">\displaystyle \frac{\displaystyle \sum_{i=1}^{c} \rho_i g_i}{\displaystyle \sum_{i=1}^{c} \rho_i}</script><span> 。</span></li><li><span>普通重要性采样（ordinary importance sampling）：</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.032ex" height="6.665ex" viewBox="0 -1621.8 4319.2 2869.7" role="img" focusable="false" 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298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E451-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E451-MJMATHI-63" x="93" y="-686"></use></g><g transform="translate(1026,0)"><use xlink:href="#E451-MJSZ2-2211" x="0" y="0"></use><g transform="translate(148,-1088)"><use transform="scale(0.707)" xlink:href="#E451-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E451-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E451-MJMAIN-31" x="1123" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E451-MJMATHI-63" x="804" y="1626"></use></g><g transform="translate(2637,0)"><use xlink:href="#E451-MJMATHI-3C1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E451-MJMATHI-69" x="731" y="-213"></use></g><g transform="translate(3498,0)"><use xlink:href="#E451-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E451-MJMATHI-69" x="674" y="-213"></use></g></g></svg></span><script type="math/tex">\displaystyle \frac{1}{c} \sum_{i=1}^{c} \rho_i g_i</script><span> 。</span></li></ul><p><span>对于加权重要性采样，如果某个权重 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.258ex" height="2.421ex" viewBox="0 -749.6 2694.5 1042.3" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E454-MJMATHI-3C1" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z"></path><path stroke-width="0" id="E454-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E454-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E454-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E454-MJMATHI-3C1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E454-MJMATHI-69" x="731" y="-213"></use><use xlink:href="#E454-MJMAIN-3D" x="1138" y="0"></use><use xlink:href="#E454-MJMAIN-30" x="2194" y="0"></use></g></svg></span><script type="math/tex">\rho_i=0</script><span> ，那么它不会让对应的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.907ex" height="1.746ex" viewBox="0 -500.4 821 751.6" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E453-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="0" id="E453-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E453-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E453-MJMATHI-69" x="674" y="-213"></use></g></svg></span><script type="math/tex">g_i</script><span> 参与平均，并不影响整体的平均值；对于普通重要性采样，如果某个权重 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.258ex" height="2.421ex" viewBox="0 -749.6 2694.5 1042.3" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E454-MJMATHI-3C1" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z"></path><path stroke-width="0" id="E454-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E454-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E454-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E454-MJMATHI-3C1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E454-MJMATHI-69" x="731" y="-213"></use><use xlink:href="#E454-MJMAIN-3D" x="1138" y="0"></use><use xlink:href="#E454-MJMAIN-30" x="2194" y="0"></use></g></svg></span><script type="math/tex">\rho_i=0</script><span> ，那么它会让 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.939ex" viewBox="0 -749.6 500 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E455-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E455-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex">0</script><span> 参与平均，使得平均值变小。（</span><em><span>两者使用时的具体区别请参考《Reinforcement Learning: An Introduction Second Edition》，Richard S,sutton 等著，第五章 5.5 节。</span></em><span>）</span></p><p><span>基于重要性采样，可以得出每次访问加权重要性采样回合更新策略评估算法 4-5 ，算法中的行为策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.996ex" height="1.939ex" viewBox="0 -749.6 429 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E456-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 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\large \textbf{算法 4-5   每次访问加权重要性采样异策回合更新评估策略的动作价值} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{1.（初始化）初始化动作价值估计 $q(s,a) \leftarrow$ 任意值，$s \in \mathcal S, a \in \mathcal A$ 。如果需要使用权重和，} \\
&\qquad \text{则初始化权重和 $c(s,a) \leftarrow 0,\; s \in \mathcal S, a \in \mathcal A$ 。} \\
&\text{2.（回合更新）对每个回合执行以下操作：} \\
&\qquad \text{2.1（行为策略）指定行为策略 $b$ ，使得 $\pi \ll b$ 。}\\
&\qquad \text{2.2（采样）用策略 $b$ 生成轨迹 $S_0,A_0,R_1,S_1,\cdots,S_{T-1},A_{T-1},R_T,S_T$ 。} \\
&\qquad \text{2.3（初始化回报和权重）$G \leftarrow 0$ ，$\rho \leftarrow 1$ 。} \\
&\qquad \text{2.4（逐步更新）对 $t \leftarrow T-1,T-2,\cdots,0$ ，执行以下操作：} \\
&\qquad \qquad \text{2.4.1（更新回报）$G \leftarrow \gamma G + R_{t+1}$ ；} \\
&\qquad \qquad \text{2.4.2（更新价值）更新 $q(S_t,A_t)$ 以减小 $\rho[G-q(S_t,A_t)]^2$ 。如 $c(S_t,A_t) \leftarrow c(S_t,A_t) + \rho$ ，} \\
&\qquad \qquad \qquad \;\, \text{$q(S_t,A_t) \leftarrow q(S_t,A_t) + \frac{\rho}{c(S_t,A_t)}[G-q(S_t,A_t)]$ 。}\\
&\qquad \qquad \text{2.4.3（更新权重）$\rho \leftarrow \rho \frac{\pi(A_t \mid S_t)}{b(A_t \mid S_t)}$ ；} \\
&\qquad \qquad \text{2.4.4（提前终止）如果 $\rho = 0$ ，则结束步骤 2.4 的循环。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>因为算法是逆序更新评估动作价值，根据其重要性采样比率表达式 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.968ex" height="2.71ex" viewBox="0 -832.7 1278 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E457-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E457-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="0" id="E457-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-eq%3A4"><rect width="1278" height="1000" y="-250" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#E457-MJMAIN-28"></use><use xlink:href="#E457-MJMAIN-34" x="389" y="0"></use><use xlink:href="#E457-MJMAIN-29" x="889" y="0"></use></g></a></g></svg></span><script type="math/tex">\eqref{eq:4}</script><span> 可知，权重值初始化为 1 ，同时根据该表达式来更新权重，如果某次权重值变为 0 ，那么之后循环的权重也都为 0 ，因此此时可以终止循环。</span></p><p><span>（</span><em><span>该算法存在的问题：考虑只有一种结局的环境，且在该环境下，存在一种策略无法到达该环境的结局，例如一个唯一出口的迷宫，如果确定性目标策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E461-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E461-MJMATHI-3C0" x="0" y="0"></use></g></svg></span><script type="math/tex">\pi</script><span> 来回打转无法走出迷宫，即无法结束环境的一个回合到达终止状态 </span><span 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47T255 29T308 24Z"></path><path stroke-width="0" id="E459-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E459-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E459-MJMATHI-54" x="866" y="-213"></use></g></svg></span><script type="math/tex">S_T</script><span> ，那么对于采样的任意轨迹都有 </span><span 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transform="scale(0.707)" xlink:href="#E460-MJMATHI-54" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E460-MJMAIN-2212" x="704" y="0"></use><use transform="scale(0.707)" xlink:href="#E460-MJMAIN-31" x="1482" y="0"></use></g></g><use xlink:href="#E460-MJMAIN-29" x="6161" y="0"></use></g><g transform="translate(132,-694)"><use xlink:href="#E460-MJMATHI-62" x="0" y="0"></use><use xlink:href="#E460-MJMAIN-28" x="429" y="0"></use><g transform="translate(818,0)"><use xlink:href="#E460-MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#E460-MJMATHI-54" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E460-MJMAIN-2212" x="704" y="0"></use><use transform="scale(0.707)" xlink:href="#E460-MJMAIN-31" x="1482" y="0"></use></g></g><use xlink:href="#E460-MJMAIN-2223" x="3347" y="0"></use><g transform="translate(3903,0)"><use xlink:href="#E460-MJMATHI-53" x="0" y="0"></use><g transform="translate(613,-150)"><use transform="scale(0.707)" xlink:href="#E460-MJMATHI-54" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E460-MJMAIN-2212" x="704" y="0"></use><use transform="scale(0.707)" xlink:href="#E460-MJMAIN-31" x="1482" y="0"></use></g></g><use xlink:href="#E460-MJMAIN-29" x="6017" y="0"></use></g></g><use xlink:href="#E460-MJMAIN-3D" x="7188" y="0"></use><use xlink:href="#E460-MJMAIN-30" x="8244" y="0"></use></g></svg></span><script type="math/tex">\displaystyle \frac{\pi(A_{T-1} \mid S_{T-1})}{b(A_{T-1} \mid S_{T-1})}=0</script><span> ，此时永远都只能更新终止状态的前一个状态。再推广到多结局环境，只要存在一种策略无法到达该环境的结局，且目标策略为这种策略，那么该算法都只能更新终止状态的前一个状态。另外该算法显然会在很多情况下提前终止，使得很多采样轨迹未被完全使用，与同策算法相同迭代次数下，更新价值的次数更少，因此可能需要更多的迭代次数才能够更好的收敛。</span></em><span>）</span></p><p><span>在评估策略动作价值的基础上加上策略改进即可得到求解最优策略的算法 4-6 。由于使用了确定性策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E461-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E461-MJMATHI-3C0" x="0" y="0"></use></g></svg></span><script type="math/tex">\pi</script><span> ，所以存在一个状态使得 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.65ex" height="2.71ex" 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385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E462-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E462-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E462-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E462-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E462-MJMAIN-28" x="573" y="0"></use><use xlink:href="#E462-MJMATHI-61" x="962" y="0"></use><use xlink:href="#E462-MJMAIN-2223" x="1768" y="0"></use><use xlink:href="#E462-MJMATHI-73" x="2324" y="0"></use><use xlink:href="#E462-MJMAIN-29" x="2793" y="0"></use><use xlink:href="#E462-MJMAIN-3D" x="3460" y="0"></use><use xlink:href="#E462-MJMAIN-31" x="4516" y="0"></use></g></svg></span><script type="math/tex">\pi(a \mid s)=1</script><span> ；若 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.051ex" height="2.71ex" viewBox="0 -832.7 4758.1 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path 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114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E463-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E463-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path><path stroke-width="0" id="E463-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E463-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E463-MJMATHI-74" x="1060" y="-213"></use><use xlink:href="#E463-MJMAIN-2260" x="1383" y="0"></use><use xlink:href="#E463-MJMATHI-3C0" x="2438" y="0"></use><use xlink:href="#E463-MJMAIN-28" x="3011" y="0"></use><g transform="translate(3400,0)"><use xlink:href="#E463-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E463-MJMATHI-74" x="866" y="-213"></use></g><use xlink:href="#E463-MJMAIN-29" x="4369" y="0"></use></g></svg></span><script type="math/tex">A_t \ne \pi(S_t)</script><span> ，意味着 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.148ex" height="2.71ex" viewBox="0 -832.7 6091.6 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E464-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E464-MJMAIN-28" d="M94 250Q94 319 104 381T127 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546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path><path stroke-width="0" id="E464-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E464-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E464-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E464-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E464-MJMAIN-28" x="573" y="0"></use><g transform="translate(962,0)"><use xlink:href="#E464-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E464-MJMATHI-74" x="1060" y="-213"></use></g><use xlink:href="#E464-MJMAIN-2223" x="2345" y="0"></use><g transform="translate(2900,0)"><use xlink:href="#E464-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E464-MJMATHI-74" x="866" y="-213"></use></g><use xlink:href="#E464-MJMAIN-29" x="3869" y="0"></use><use xlink:href="#E464-MJMAIN-3D" x="4535" y="0"></use><use xlink:href="#E464-MJMAIN-30" x="5591" y="0"></use></g></svg></span><script type="math/tex">\pi(A_t \mid S_t)=0</script><span> ，否则为  </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.148ex" height="2.71ex" viewBox="0 -832.7 6091.6 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E465-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E465-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 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240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E465-MJMAIN-2223" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E465-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E465-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E465-MJMAIN-28" x="573" y="0"></use><g transform="translate(962,0)"><use xlink:href="#E465-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E465-MJMATHI-74" x="1060" y="-213"></use></g><use xlink:href="#E465-MJMAIN-2223" x="2345" y="0"></use><g transform="translate(2900,0)"><use xlink:href="#E465-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E465-MJMATHI-74" x="866" y="-213"></use></g><use xlink:href="#E465-MJMAIN-29" x="3869" y="0"></use><use xlink:href="#E465-MJMAIN-3D" x="4535" y="0"></use><use xlink:href="#E465-MJMAIN-31" x="5591" y="0"></use></g></svg></span><script type="math/tex">\pi(A_t \mid S_t)=1</script><span>  ；据此就可以得到步骤 2.4.4 和 2.4.5 了。对该算法稍加修改，即可得到首次访问的算法和普通重要性采样的算法，此处略。</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block 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y="0"></use><g transform="translate(818,0)"><use xlink:href="#E403-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E403-MJMATHI-74" x="1060" y="-213"></use></g><use xlink:href="#E403-MJMAIN-2223" x="2201" y="0"></use><g transform="translate(2756,0)"><use xlink:href="#E403-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E403-MJMATHI-74" x="866" y="-213"></use></g><use xlink:href="#E403-MJMAIN-29" x="3725" y="0"></use></g></g></g></g><g transform="translate(14103,0)"><g transform="translate(250,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">。</text></g></g></g></g><g transform="translate(0,-9844)"><g><rect fill="black" stroke="none" width="41597" height="100" x="0" y="-500"></rect></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-401">\; \\ \; \\
\large \textbf{算法 4-6   每次访问加权重要性采样异策回合更新最优策略求解} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{1.（初始化）初始化动作价值估计 $q(s,a) \leftarrow$ 任意值，$s \in \mathcal S, a \in \mathcal A$ 。如果需要使用权重和，} \\
&\qquad \text{则初始化权重和 $c(s,a) \leftarrow 0,\; s \in \mathcal S, a \in \mathcal A$ ；$\pi(s) \leftarrow \underset{a}{\arg\max}\; q(s,a), \; s \in \mathcal S$ 。} \\
&\text{2.（回合更新）对每个回合执行以下操作：} \\
&\qquad \text{2.1（柔性策略）指定 $b$ 为任意柔性策略。}\\
&\qquad \text{2.2（采样）用策略 $b$ 生成轨迹 $S_0,A_0,R_1,S_1,\cdots,S_{T-1},A_{T-1},R_T,S_T$ 。} \\
&\qquad \text{2.3（初始化回报和权重）$G \leftarrow 0$ ，$\rho \leftarrow 1$ 。} \\
&\qquad \text{2.4（逐步更新）对 $t \leftarrow T-1,T-2,\cdots,0$ ，执行以下操作：} \\
&\qquad \qquad \text{2.4.1（更新回报）$G \leftarrow \gamma G + R_{t+1}$ ；} \\
&\qquad \qquad \text{2.4.2（更新价值）更新 $q(S_t,A_t)$ 以减小 $\rho[G-q(S_t,A_t)]^2$ ；} \\
&\qquad \qquad \text{2.4.3（策略更新）$\pi(S_t) \leftarrow \underset{a}{\arg\max}\; q(S_t,a)$ ；} \\
&\qquad \qquad \text{2.4.4（提前终止）若 $A_t \ne \pi(S_t)$ 则退出步骤 2.4 ；} \\
&\qquad \qquad \text{2.4.5（更新权重）$\rho \leftarrow \rho \frac{1}{b(A_t \mid S_t)}$ 。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>（</span><em><span>该算法存在与算法 4-5 类似的问题：假设环境为一条从左到右拥有 n 个状态的直线迷宫，那么只有在终点前一个状态时往右走，才能到达终点走出迷宫结束回合。设到达终点奖励为 10 ，记为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.718ex" height="2.71ex" viewBox="0 -832.7 2892.3 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E466-MJMATHI-52" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></path><path stroke-width="0" id="E466-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E466-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 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xmlns:xlink="http://www.w3.org/1999/xlink" width="5.535ex" height="2.71ex" viewBox="0 -832.7 2383.3 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E468-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E468-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 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& G = 0 + R_T(10) = 10 \\
& c(S(n-1), A(2)) = 0 + 1 \\
& q(S(n-1), A(2)) = 20 - \frac{1}{1} \times (10 - 20) = 10 \\
& \pi(S(n-1)) = \underset{a}{\arg\max}\; q(S(n-1), a) = A(1) \ne A(2)
\end{split}</script></div></div><p><em><span>只使用了最后一次动作轨迹就直接结束了，然后对于第二回合的任意轨迹，其最后一次动作轨迹依旧相同，那么无论循环多少次，改变的只有权重和 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.255ex" height="2.71ex" viewBox="0 -832.7 7429.1 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E476-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path><path stroke-width="0" id="E476-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 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transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E480-MJMATHI-71" x="0" y="0"></use><use xlink:href="#E480-MJMAIN-28" x="460" y="0"></use><use xlink:href="#E480-MJMATHI-53" x="849" y="0"></use><use xlink:href="#E480-MJMAIN-28" x="1494" y="0"></use><use xlink:href="#E480-MJMATHI-6E" x="1883" y="0"></use><use xlink:href="#E480-MJMAIN-2212" x="2705" y="0"></use><use xlink:href="#E480-MJMAIN-31" x="3705" y="0"></use><use xlink:href="#E480-MJMAIN-29" x="4205" y="0"></use><use xlink:href="#E480-MJMAIN-2C" x="4594" y="0"></use><use xlink:href="#E480-MJMATHI-41" x="5039" y="0"></use><use xlink:href="#E480-MJMAIN-28" x="5789" y="0"></use><use xlink:href="#E480-MJMAIN-32" x="6178" y="0"></use><use xlink:href="#E480-MJMAIN-29" x="6678" y="0"></use><use xlink:href="#E480-MJMAIN-29" x="7067" y="0"></use></g></svg></span><script type="math/tex">q(S(n-1), A(2))</script><span> 的值时，逐步更新循环才能进行第二次循环，在对于第二次循环的过程中，也存在类似的问题。</span></em><span>）</span></p><h3><a name="三案例21-点游戏blackjack-v0）" class="md-header-anchor"></a><span>三、案例：21 点游戏（Blackjack-v0）</span></h3><p><span>使用 gym 库中的纸牌游戏 “21点”（Blackjack-v0）作为案例分析，为其实现游戏 AI。该游戏规则可百度或者查看</span><a href='https://github.com/openai/gym/blob/master/gym/envs/toy_text/blackjack.py' target='_blank' title='Blackjack-v0'><span>源代码</span></a><span>，另外使用学习代码中的 </span><code>run_episode</code><span> 函数也能够助于理解该游戏过程。</span></p><p><span>由于该游戏具有以下特点：</span></p><ul><li><span>在一个轨迹中不可能出现重复的状态，因此不需要区分首次访问和每次访问。</span></li><li><span>在一个轨迹中只有最后的一个奖励值是非零值。</span></li><li><span>（</span><em><span>游戏过程不会涉及算法 4-5 和 4-6 所存在的问题，实现算法时不做额外考虑。</span></em><span>）</span></li></ul><p><span>因此当限定 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.52ex" height="2.421ex" viewBox="0 -749.6 2376.6 1042.3" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E481-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 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-21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></path><path stroke-width="0" id="E482-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E482-MJMATHI-47" x="0" y="0"></use><use xlink:href="#E482-MJMAIN-3D" x="1063" y="0"></use><g transform="translate(2119,0)"><use xlink:href="#E482-MJMATHI-52" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E482-MJMATHI-54" x="1073" y="-213"></use></g></g></svg></span><script type="math/tex">G=R_T</script><span> ，且在更新时能够进行顺序更新。那么采样轨迹只需要记录状态和动作以及最后一个奖励值，可将书中源代码的采样轨迹部分提取，作为一个单独的用来采样轨迹的函数 </span><code>explore_sampling</code><span> ，并返回需要记录的值即可：</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 43px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 39px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>11</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -39px; width: 39px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">explore_sampling</span>(<span class="cm-variable">env</span>, <span class="cm-variable">policy</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">state_actions</span> = []</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">state</span> = <span class="cm-variable">env</span>.<span class="cm-property">reset</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">while</span> <span class="cm-keyword">True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state</span> = (<span class="cm-variable">state</span>[<span class="cm-number">0</span>], <span class="cm-variable">state</span>[<span class="cm-number">1</span>], <span class="cm-builtin">int</span>(<span class="cm-variable">state</span>[<span class="cm-number">2</span>]))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">action</span> = <span class="cm-variable">np</span>.<span class="cm-property">random</span>.<span class="cm-property">choice</span>(<span class="cm-variable">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>, <span class="cm-variable">p</span>=<span class="cm-variable">policy</span>[<span class="cm-variable">state</span>])</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state_actions</span>.<span class="cm-property">append</span>((<span class="cm-variable">state</span>, <span class="cm-variable">action</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state</span>, <span class="cm-variable">reward</span>, <span class="cm-variable">done</span>, <span class="cm-variable">_</span> = <span class="cm-variable">env</span>.<span class="cm-property">step</span>(<span class="cm-variable">action</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">done</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">break</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">return</span> <span class="cm-variable">state_actions</span>, <span class="cm-variable">reward</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 352px;"></div><div class="CodeMirror-gutters" style="height: 352px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 38px;"></div></div></div></div></pre><p><span>然后在实现算法时，采样部分改为调用该函数即可。例如使用算法 4-1 实现策略评估：</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 43px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 39px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -39px; width: 39px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">evaluate_action_monte_carlo</span>(<span class="cm-variable">env</span>, <span class="cm-variable">policy</span>, <span class="cm-variable">episode_num</span>=<span class="cm-number">500000</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">q</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros_like</span>(<span class="cm-variable">policy</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">c</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros_like</span>(<span class="cm-variable">policy</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">_</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">episode_num</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state_actions</span>, <span class="cm-variable">reward</span> = <span class="cm-variable">explore_sampling</span>(<span class="cm-variable">env</span>, <span class="cm-variable">policy</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">g</span> = <span class="cm-variable">reward</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">state</span>, <span class="cm-variable">action</span> <span class="cm-keyword">in</span> <span class="cm-variable">state_actions</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">c</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>] += <span class="cm-number">1.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>] += (<span class="cm-variable">g</span> <span class="cm-operator">-</span> <span class="cm-variable">q</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>]) <span class="cm-operator">/</span> <span class="cm-variable">c</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">return</span> <span class="cm-variable">q</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 320px;"></div><div class="CodeMirror-gutters" style="height: 320px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 38px;"></div></div></div></div></pre><p><span>根据该游戏规则，可以知道当玩家点数小于等于 11 点时，再要一张牌肯定不会超过 21 点，因此玩家点数在 12~21 点是更加值得去关注的状态。下面使用算法 4-3 来实现玩家起始点数为 12~21 点的最优策略求解，由于该环境下相同总点数的玩家持牌都是等价的，因此只需任意指定一种玩家的持牌即可：</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 43px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 39px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>36</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -39px; width: 39px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">monte_carlo_with_exploring_start</span>(<span class="cm-variable">env</span>, <span class="cm-variable">episode_num</span>=<span class="cm-number">500000</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">policy</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>((<span class="cm-number">22</span>, <span class="cm-number">11</span>, <span class="cm-number">2</span>, <span class="cm-number">2</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">policy</span>[:, :, :, <span class="cm-number">1</span>] = <span class="cm-number">1.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">q</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros_like</span>(<span class="cm-variable">policy</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">c</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros_like</span>(<span class="cm-variable">policy</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">_</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">episode_num</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state</span> = (<span class="cm-variable">np</span>.<span class="cm-property">random</span>.<span class="cm-property">randint</span>(<span class="cm-number">12</span>, <span class="cm-number">22</span>), <span class="cm-error">\</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">np</span>.<span class="cm-property">random</span>.<span class="cm-property">randint</span>(<span class="cm-number">1</span>, <span class="cm-number">11</span>), <span class="cm-error">\</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">np</span>.<span class="cm-property">random</span>.<span class="cm-property">randint</span>(<span class="cm-number">2</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">action</span> = <span class="cm-variable">np</span>.<span class="cm-property">random</span>.<span class="cm-property">randint</span>(<span class="cm-number">2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">env</span>.<span class="cm-property">reset</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">state</span>[<span class="cm-number">2</span>]:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">env</span>.<span class="cm-property">player</span> = [<span class="cm-number">1</span>, <span class="cm-variable">state</span>[<span class="cm-number">0</span>] <span class="cm-operator">-</span> <span class="cm-number">11</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">state</span>[<span class="cm-number">0</span>] == <span class="cm-number">21</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">env</span>.<span class="cm-property">player</span> = [<span class="cm-number">10</span>, <span class="cm-number">9</span>, <span class="cm-number">2</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">env</span>.<span class="cm-property">player</span> = [<span class="cm-number">10</span>, <span class="cm-variable">state</span>[<span class="cm-number">0</span>] <span class="cm-operator">-</span> <span class="cm-number">10</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">env</span>.<span class="cm-property">dealer</span>[<span class="cm-number">0</span>] = <span class="cm-variable">state</span>[<span class="cm-number">1</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state_actions</span> = []</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">while</span> <span class="cm-keyword">True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state_actions</span>.<span class="cm-property">append</span>((<span class="cm-variable">state</span>, <span class="cm-variable">action</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state</span>, <span class="cm-variable">reward</span>, <span class="cm-variable">done</span>, <span class="cm-variable">_</span> = <span class="cm-variable">env</span>.<span class="cm-property">step</span>(<span class="cm-variable">action</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">done</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">break</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">state</span> = (<span class="cm-variable">state</span>[<span class="cm-number">0</span>], <span class="cm-variable">state</span>[<span class="cm-number">1</span>], <span class="cm-builtin">int</span>(<span class="cm-variable">state</span>[<span class="cm-number">2</span>]))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">action</span> = <span class="cm-variable">np</span>.<span class="cm-property">random</span>.<span class="cm-property">choice</span>(<span class="cm-variable">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>, <span class="cm-variable">p</span>=<span class="cm-variable">policy</span>[<span class="cm-variable">state</span>])</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">g</span> = <span class="cm-variable">reward</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">state</span>, <span class="cm-variable">action</span> <span class="cm-keyword">in</span> <span class="cm-variable">state_actions</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">c</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>] += <span class="cm-number">1.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>] += (<span class="cm-variable">g</span> <span class="cm-operator">-</span> <span class="cm-variable">q</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>]) <span class="cm-operator">/</span> <span class="cm-variable">c</span>[<span class="cm-variable">state</span>][<span class="cm-variable">action</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">a</span> = <span class="cm-variable">q</span>[<span class="cm-variable">state</span>].<span class="cm-property">argmax</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">policy</span>[<span class="cm-variable">state</span>] = <span class="cm-number">0.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">policy</span>[<span class="cm-variable">state</span>][<span class="cm-variable">a</span>] = <span class="cm-number">1.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">return</span> <span class="cm-variable">policy</span>, <span class="cm-variable">q</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1152px;"></div><div class="CodeMirror-gutters" style="height: 1152px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 38px;"></div></div></div></div></pre><p><span>其他的最优策略求解算法也是类似的，这里不再展示代码，具体可查看代码目录下的 </span><code>4.Blackjack-v0.py</code><span> 。在得到最优策略后可对最优策略进行测试，函数 </span><code>test_policy</code><span> 在指定回合数下与庄家进行游戏，并统计输赢和平局回合数。</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 43px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 39px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>12</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -39px; width: 39px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">test_policy</span>(<span class="cm-variable">env</span>, <span class="cm-variable">policy</span>, <span class="cm-variable">episode_num</span>=<span class="cm-number">50000</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">win</span>, <span class="cm-variable">tied</span>, <span class="cm-variable">lost</span> = <span class="cm-number">0</span>, <span class="cm-number">0</span>, <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">_</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">episode_num</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">score</span> = <span class="cm-variable">run_episode</span>(<span class="cm-variable">env</span>, <span class="cm-variable">policy</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">score</span> == <span class="cm-number">1</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">win</span> += <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">elif</span> <span class="cm-variable">score</span> == <span class="cm-number">0</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">tied</span> += <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">elif</span> <span class="cm-variable">score</span> == <span class="cm-operator">-</span><span class="cm-number">1</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">lost</span> += <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-builtin">print</span>(<span class="cm-string">"AI win {} round, tied {} round, lost {} round, in {} round, against dealer."</span><span class="cm-error">\</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  .<span class="cm-property">format</span>(<span class="cm-variable">win</span>, <span class="cm-variable">tied</span>, <span class="cm-variable">lost</span>, <span class="cm-variable">episode_num</span>))</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 384px;"></div><div class="CodeMirror-gutters" style="height: 384px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 38px;"></div></div></div></div></pre><p>&nbsp;</p></div>
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